Interesting behavior of lm() with small, problematic data sets

I've recently come across the following results reported from the lm() function when applied to a particular type of admittedly difficult data.  When working with
small data sets (for instance 3 points) with the same response for different predicting variable, the resulting slope estimate is a reasonable approximation of the expected 0.0, but the p-value of that slope estimate is a surprising value.  A reproducible example is included below, along with the output of the summary of results

######### example code
x <- c(1,2,3)
y <- c(1,1,1)

#above results in{ (1,1) (2,1) (3,1)} data set to regress

new.rez <- lm (y ~ x) # regress constant y on changing x)
summary(new.rez) # display results of regression

######## end of example code

Results:

Call:
lm(formula = y ~ x)

Residuals:
         1          2          3
 5.906e-17 -1.181e-16  5.906e-17

Coefficients:
              Estimate Std. Error    t value Pr(>|t|)
(Intercept)  1.000e+00  2.210e-16  4.525e+15   <2e-16 ***
x           -1.772e-16  1.023e-16 -1.732e+00    0.333
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1

Residual standard error: 1.447e-16 on 1 degrees of freedom
Multiple R-squared:  0.7794,    Adjusted R-squared:  0.5589
F-statistic: 3.534 on 1 and 1 DF,  p-value: 0.3112

Warning message:
In summary.lm(new.rez) : essentially perfect fit: summary may be unreliable

##############

There is a warning that the summary may be unreliable sue to the essentially perfect fit, but a p-value of 0.3112 doesn?t seem reasonable.
As a side note, the various r^2 values seem odd too.

Tim Glover
Senior Scientist II (Geochemistry, Statistics), Americas - Environment & Infrastructure, Amec Foster Wheeler
271 Mill Road, Chelmsford, Massachusetts, USA 01824-4105
T +01 978 692 9090      D +01 978 392 5383      M +01 850 445 5039
tim.glover at amecfw.com      amecfw.com

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